A vortex ring is an
axially symmetric solution of (REF ) which does not change shape in time, whose
vorticity is mostly concentrated inside a solid torus which moves with constant speed along
the symmetry axis.
The vortex lines form large circles that fill the torus, whereas fluid particles
spin around the vortex core within perpendicular cross sections characterized with a thin torus‐shaped region in which
the vorticity of the fluid is concentrated.
These objects were first described by Helmholtz
in his celebrated work [1]}, [2]}. He considered with great
attention the situation where the vorticity field is concentrated in a circular vortex-filament
of very small section, a thin vortex ring.
Helmholtz also analyzed the interaction between two or more similar coaxial vortex rings with thin sections
and similar translation speeds.
As pointed out by Jerrard and Smets [3]}, his description reads:
